The clissification of the Norwegian language has seen more than its fair share of discussion, especially in Norway, of course. But one has to remember that such classification must be based on written languages, and the most common written form of Norwegian (Bokmål) is indeed based on Danish. It has been proposed that the other written form, Nynorsk, should be classed with the Western languages, but even though it is based on Western and Northern dialects, its grammar and syntax is quite clearly Eastern Nordic.
"Must"? Says who?
I certainly didn't.
Also, while the tradition of Riksmål and Bokmål goes back to Danish, describing it as "based on Danish" is omitting the reforms that have been made since the 1800s, to bring them closer to Norwegian spoken language. And since Nynorsk was designed using the same orthography, you could as easily say it too is "based on Danish". (Drawing the line between those written forms seems strange to me: They are far more similar to one another than they are to either Icelandic or Danish. I rather suspect those who suggest different classifications of the written Norwegian languages of having an agenda in which the similarity of those languages are ... inconvenient.)
Also also: Grammar and syntax that is clearly Eastern Nordic? Only if you ignore the differences between it and the certainly Eastern Nordic language (trust me: there are differences), or accept them as Eastern Nordic per definition. Which seems to make the question moot ...
Bit of a rant, I suppose:
The difference in grammar and syntax between spoken Norwegian and Icelandic is mostly due to the great "simplifications" that Norwegian saw, often attributed to the Hanseatic League and other trade connections, and somewhat parallelling the "simplification" that produced Modern English from Old English (or modern Swedish and Danish for that matter). Icelandic, starting from the same language as Norwegian, developed in another direction, with less "simplification".
As the Hanseatic League declined, Danish was adopted (also) as trade language in Norway, becoming pretty much the only written language of Norway (until the reforms of Riksmål and invention of Nynorsk). But at that time, the spoken language had already come most of the way from Old Norse to modern spoken Norwegian.
Most Norwegians take great pride in their spoken language. And written Norwegian is considered Eastern Nordic only to the extent that it is considered foreign. Many do speak of them as Danish. (Or some of them. I don't approve of drawing a line between them.)
I spoke (wrote) of Norwegian. If Bokmål is Danish, Norwegian is a spoken language.